Matrix Computational Assumptions in Multilinear Groups.

We put forward a new family of computational assumptions, the Kernel Matrix DiffieHellman Assumption. Given some matrix A sampled from some distribution D`,k, the kernel assumption says that it is hard to find “in the exponent” a nonzero vector in the kernel of Au003e. This family is the natural computational analogue of the Matrix Decisional Diffie-Hellman Assumption (MDDH), proposed by Escala et al. As such it allows to extend the advantages of their algebraic framework to computational assumptions. The k-Decisional Linear Assumption is an example of a family of decisional assumptions of strictly increasing hardness when k grows. We show that for any such family the corresponding Kernel Assumption family is also a strictly increasingly weaker family of computational assumptions. This requires ruling out the existence of some black-box reductions between flexible problems (i.e., computational problems with a non unique solution).